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Proving the Limit at a Point of a Function

Recall from The Limit of a Function page that if $f : A \to \mathbb{R}$ and $c$ is a cluster point of $A$ then $\lim_{x \to c} f(x) = L$ if $\forall \epsilon > 0$$\exists \delta > 0$ such that if $x \in A$ and $0 < \mid x - c \mid < \delta$ then $\mid f(x) - L \mid < \epsilon$. We will now use this formal definition of a limit of a function at a point $c$ in actual proofs.